标题： 闭子局部的并集不总是余框架 显示英文标题

标题： Joins of closed sublocales are not always a coframe

摘要： 给定一个局部 $L$，闭子局部的并集的集合 $\mathsf{S}_c(L)$形成一个框架——有些出乎意料，因为它自然地嵌入到所有局部 $L$的余框架中，其中我们所说的余框架是指框架的序理论对偶。 这一构造在无点拓扑中引起了关注：作为框架范畴中的极大本质扩张，因其（非）函子性质，它与框架的规范扩张和精确滤子的关系等。 该理论的一个中心开放问题是由 Picado、Pultr 和 Tozzi 在 2019 年提出的，他们询问 $\mathsf{S}_c(L)$是否总是余框架，或者是否存在一个局部使得此性质不成立。 在本文中，我们通过构造一个局部 $L$使得 $\mathsf{S}_c(L)$不是余框架，从而否定了这个问题。 此类问题的主要挑战在于理解$\mathsf{S}_c(L)$中的精确下确界的难度；我们通过分析$\mathsf{S}_c(L)$满足的某种分离性质来规避这一问题。 显示英文摘要

摘要： Given a locale $L$, the collection $\mathsf{S}_c(L)$ of joins of closed sublocales forms a frame--somewhat unexpectedly, as it is naturally embedded in the coframe of all sublocales of $L$, where by coframe we mean the order-theoretic dual of a frame. This construction has attracted attention in point-free topology: as a maximal essential extension in the category of frames, for its (non-)functorial properties, its relation to canonical extensions and exact filters of frames, etc. A central open question of the theory, posed by Picado, Pultr, and Tozzi in 2019, asked whether $\mathsf{S}_c(L)$ is always a coframe, or whether there exists a locale for which this fails. In this paper, we resolve this question in the negative by constructing a locale $L$ such that $\mathsf{S}_c(L)$ is not a coframe. The main challenge in such questions lies in the difficulty of understanding exact infima in $\mathsf{S}_c(L)$; we circumvent this by analysing a certain separation property satisfied by $\mathsf{S}_c(L)$.